Mikhail I. Ostrovskii

Bio: Mikhail I. Ostrovskii is an academic researcher from St. John's University. The author has contributed to research in topics: Banach space & Banach manifold. The author has an hindex of 14, co-authored 121 publications receiving 834 citations. Previous affiliations of Mikhail I. Ostrovskii include The Catholic University of America & University of California, Riverside.

Metric Embeddings: Bilipschitz and Coarse Embeddings Into Banach Spaces

Abstract: Embeddings of discrete metric spaces into Banach spaces recently became an important tool in computer science and topology. The book will help readers to enter and to work in this very rapidly developing area having many important connections with different parts of mathematics and computer science. The purpose of the book is to present some of the most important techniques and results, mostly on bilipschitz and coarse embeddings. The topics include embeddability of locally finite metric spaces into Banach spaces is finitely determined, constructions of embeddings, distortion in terms of Poincare inequalities, constructions of families of expanders and of families of graphs with unbounded girth and lower bounds on average degrees, Banach spaces which do not admit coarse embeddings of expanders, structure of metric spaces which are not coarsely embeddable into a Hilbert space, applications of Markov chains to embeddability problem, metric characterizations of properties of Banach spaces, and Lipschitz free spaces.

Distances between Banach spaces

Abstract: The main object of the paper is to study the distance between Banach spaces introduced by Kadets. For Banach spaces X and Y, the Kadets distance is dened to be the inmum of the Hausdor distance d(B X , B Y ) between the respective closed unit balls over all isometric linear embeddings of X and Y into a common Banach space Z. This is compared with the Gromov-Hausdor distance which is dened to be the inmum of d(B X , B Y ) over all isometric embeddings into a common metric space Z. We prove continuity type results for the Kadets distance including a result that shows that this notion of distance has appli- cations to the theory of complex interpolation. 1991 Mathematics Subject Classication: 46B20, 46M35; 46B03, 54E35.

Sobolev spaces on graphs

Abstract: The present paper is devoted to discrete analogues of Sobolev spaces of smooth functions. The discrete analogues that we consider are spaces of functions on vertex sets of graphs. Such spaces have applications in Graph Theory, Metric Geometry and Convex Geometry. We present known and prove some new results on Banach-space-theoretical properties of such spaces. Keywords: Sobolev space on a graph; isoperimetric constant of a graph; Cheeger constant; Banach-Mazur distance Quaestiones Mathematicae 28(2005), 501–523.

Topologies on the set of all subspaces of a banach space and related questions of banach space geometry

Abstract: The survey is devoted to two of the most common topologies on the set of all subspaces of a Banach space. The first part contains definitions of the topologies and a description of their properties. The main topics of the rest of the paper are: (1) Comparison of the introduced topologies; (2) Stability of properties of subspaces with respect to the topologies. The last section contains a list of open problems

Gromov–Wasserstein Distances and the Metric Approach to Object Matching

Abstract: This paper discusses certain modifications of the ideas concerning the Gromov–Hausdorff distance which have the goal of modeling and tackling the practical problems of object matching and comparison. Objects are viewed as metric measure spaces, and based on ideas from mass transportation, a Gromov–Wasserstein type of distance between objects is defined. This reformulation yields a distance between objects which is more amenable to practical computations but retains all the desirable theoretical underpinnings. The theoretical properties of this new notion of distance are studied, and it is established that it provides a strict metric on the collection of isomorphism classes of metric measure spaces. Furthermore, the topology generated by this metric is studied, and sufficient conditions for the pre-compactness of families of metric measure spaces are identified. A second goal of this paper is to establish links to several other practical methods proposed in the literature for comparing/matching shapes in precise terms. This is done by proving explicit lower bounds for the proposed distance that involve many of the invariants previously reported by researchers. These lower bounds can be computed in polynomial time. The numerical implementations of the ideas are discussed and computational examples are presented.

Metric Spaces Of Non Positive Curvature

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Determinants and matrices. By A. C. Aitken. Pp. vii, 135. 4s. 6d. 1939. University Mathematical Texts, 1. (Oliver and Boyd)

Abstract: $$\\left[ {\\begin{array}{*{20}{c}} {10} \\\\ { - v{{a}^{{ - 1}}}1} \\\\ \\end{array} } \\right]\\left[ {\\begin{array}{*{20}{c}} {au} \\\\ {vb} \\\\ \\end{array} } \\right]\\left[ {\\begin{array}{*{20}{c}} {1 - {{a}^{{ - 1}}}u} \\\\ {01} \\\\ \\end{array} } \\right] = \\left[ {\\begin{array}{*{20}{c}} {a0} \\\\ {0b - v{{a}^{{ - 1}}}u} \\\\ \\end{array} } \\right]$$ (1.1) provided a is regular.